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# Logically equivalent

Two sentences are Logically Equivalent if and only if they are true in precisely the same situations. Another way to think of a pair of logically equivalent sentences is as two sentences that say the exact same thing. If one of them is true, the other is true; if one of them is false, the other is false.

The notion of logical equivalence can be broadened to apply to collections of sentences of any size (not just pairs).

## Truth Table Test for Logical Equivalence

The easiest way to test whether two sentences are logically equivalent is to make a truth-table for the sentences in question. Once you fill out the truth-table for the sentences, you look at the column under the main connective of each sentence. The two sentences are logically equivalent exactly when the column under the main connective of the first sentence is identical to the column under the main connective of the second sentence. Notice that this captures in a formal way the informal idea that two sentences are logically equivalent just in case they are either both true or both false in any given situation.

## Examples

The sentences $¬\left(A\vee B\right)$ and $\left(¬A\wedge ¬B\right)$ are logically equivalent, as the following truth-table shows:

A B $¬\left(A\vee B\right)$ $\left(¬A\wedge ¬B\right)$
T T F F
T F F F
F T F F
F F T T

The sentences $¬\left(A\wedge B\right)$ and $\left(¬A\vee ¬B\right)$ are also logically equivalent, as the following truth-table shows:

A B $¬\left(A\wedge B\right)$ $\left(¬A\vee ¬B\right)$
T T F F
T F T T
F T T T
F F T T

The sentences $\left(P\to Q\right)$ and $\left(Q\to P\right)$ are not logically equivalent, as the following truth-table demonstrates:

P Q $\left(P\to Q\right)$ $\left(Q\to P\right)$
T T T T
T F F T
F T T F
F F T T